PHYSICAL REVIEW E 85, 056602 (2012)

Electromagnetic forces on a discrete spherical invisibility cloak under time-harmonic illumination

Patrick C. Chaumet,1 Adel Rahmani,2 Frederic Zolla,1 and Andre Nicolet11Institut Fresnel, CNRS, Aix-Marseille Universite, Campus de St-Jerome 13013 Marseille, France

2School of Mathematical Sciences, University of Technology, Sydney, Broadway NSW 2007, Australia(Received 15 November 2011; published 8 May 2012)

We study electromagnetic forces and torques on a discrete spherical invisibility cloak under time-harmonicillumination. We consider the influence of material absorption and losses, and we show that while the impactof absorption on the optical force remains confined to frequencies near the absorption peak, its impact on theelectromagnetic torque experienced by the cloak is spectrally broader and follows the spectrum of the absorptioncross section of the cloak. We also investigate the mechanical shielding of a test particle within the cloak. Wefind that even an imperfect cloak can reduce the radiation pressure on the particle significantly; however, undercertain conditions the force on the particle can be stronger than it would be in the absence of the cloak.

DOI: 10.1103/PhysRevE.85.056602 PACS number(s): 41.20.−q, 42.50.Wk, 02.70.−c

I. INTRODUCTION

Recent studies have focused on the electromagnetic forcesexperienced by an electromagnetic cloak in either the fre-quency domain (time-harmonic regime) or the time domain.For the time-harmonic case, it has been shown that the opticalforce on an ideal electromagnetic cloak is exactly zero [1,2]and that the cloak remains at rest. On the other hand, in thetime domain, an electromagnetic cloak, initially at rest andilluminated by an electromagnetic pulse, would experiencea nonzero electromagnetic force and hence move while itinteracts with the pulse. By the time the pulse has died out,however, the cloak would have acquired a zero net momentumand experienced a zero net displacement [3].

In this article we deal with the time-harmonic regime, andwe study how a nonideal cloak interacts optomechanically withan electromagnetic wave. Any actual cloaking device, even inthe absence of fabrication imperfections, can achieve onlypartial cloaking due to two main factors: material dispersionand absorption, and the composite (i.e., discrete) geometry ofthe metamaterial forming the cloak. Although the effects ofdispersion and absorption are sometimes taken into accountin the definition of the permittivity and permeability of thecloak, the discrete geometry is more difficult to account forand therefore seldom considered. Our goal is to study theeffect of the composite geometry of the cloak on opticalforces, without having to worry about the specifics of themetamaterial implementation used to describe the cloak. Tothis end we use a generic model of a discrete (i.e., composite)cloak based on the discrete dipole approximation (DDA) [4–6].In the DDA formalism, the computation of the electromagneticforce on a discrete cloak amounts to computing the forceon a collection of anisotropic magnetodielectric particles.Recent studies have addressed optical forces on isotropicmagnetodielectric particles [7–9]. In this article we start bybriefly describing, in Sec. II, how particles with tensorialpermittivity and permeability can be handled in optical forcecalculations. In Sec. III we present the results obtainedfor a discretized cloak. We also discuss the influence ofmaterial dispersion and absorption on the electromagneticforce experienced by the cloak, and we investigate how wella particle inside the cloak is protected from optical forces. InSec. IV we present our conclusion.

II. OPTICAL FORCES AND TORQUES ON ANANISOTROPIC, MAGNETODIELECTRIC OBJECT

Consider an object whose electromagnetic properties aredescribed by position-dependent permittivity and permeabilitytensors. We discretize the object into a set of N polarizablesubunits over a cubic lattice with period d [10,11]; i.e.,d is the mesh size for the discretization. Each subunit ischaracterized by an electric polarizability tensor αe anda magnetic polarizability tensor αm. When the object isilluminated by an incident electromagnetic field (Einc,Hinc)the local fields at a subunit located at ri can then be written as

E(ri ,ω) = Einc(ri ,ω) +N∑

j=1

[Tee(ri ,rj ,ω)αe(rj ,ω)E(rj ,ω)

+ Tem(ri ,rj ,ω)αm(rj ,ω)H(rj ,ω)], (1)

H(ri ,ω) = Hinc(ri ,ω) +N∑

j=1

[Tme(ri ,rj ,ω)αe(rj ,ω)E(rj ,ω)

+ Tmm(ri ,rj ,ω)αm(rj ,ω)H(rj ,ω)], (2)

where ω is the angular frequency, and the quantities labeled Tare free-space field susceptibility tensors [7,12]. For the sakeof brevity we will omit the dependence on ω henceforth. Thepolarizabilities of the subunits, in the anisotropic case, aredefined from the Clausius-Mossotti relation:

αe0(ri) = 3d3

4π[ε(ri) − I][ε(ri) + 2I]−1, (3)

αm0 (ri) = 3d3

4π[μ(ri) − I][μ(ri) + 2I]−1. (4)

Taking radiation reaction [5] into account and introducing thewave number k0, the polarizabilities tensors in Eqs. (1) and (2)can be written as

αe(rj ) = [I − (2/3)ik3

0αe0(ri)

]−1αe

0(ri), (5)

αm(rj ) = [I − (2/3)ik3

0αm0 (ri)

]−1αm

0 (ri). (6)

Ultimately, the scattering problem is cast as a 6N × 6N linearsystem, which can be solved for the electric and magneticfields inside the object. For large objects (with respect to thewavelength), the linear system should be solved iteratively

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CHAUMET, RAHMANI, ZOLLA, AND NICOLET PHYSICAL REVIEW E 85, 056602 (2012)

using a solver adapted to the magnetodielectric nature of theobject [13]. Once the fields inside the object are known, thefields anywhere outside the object can be calculated by addingthe contributions of all the subunits to the scattered field. Sincewe are not merely interested in the scattered fields but in theelectromagnetic force and torque as well, we also need thespatial derivatives of the fields. These spatial derivatives atsubunit i are obtained as [14]

∇E(ri) = ∇Einc(ri) +N∑

j=1

[∇Tee(ri ,rj )αe(rj )E(rj )

+∇Tem(ri ,rj )αm(rj )H(rj )], (7)

∇H(ri) = ∇Hinc(ri) +N∑

j=1

[∇Tme(ri ,rj )αe(ri)E(ri)

+∇Tmm(ri ,rj )αm(rj )H(rj )]. (8)

Once the local field and their spatial derivatives are known ateach subunit, the kth Cartesian component of the optical forceexperienced by the ith subunit of the discretized object can bewritten as (repeated indices are summed over) [7]

Fk(ri) = 1

2Re

{pl(ri)∂

k[El(ri)]∗ + ml∂k[Hl(ri)]

∗

− 2k40

3εklnpl(ri)[m

n(ri)]∗}, (9)

where εkln is the Levi-Civita tensor, k, l, or n stands for eitherx, y, or z, and * denotes the complex conjugate of a complexvariable. As discussed in Ref. [7] the third term in Eq. (9)becomes negligible if the object under study is discretized insmall enough subunits.

To compute the optical torque we must add up the intrinsicand extrinsic torques experienced by each subunits. However,as emphasized in Refs. [15–17], to satisfy the conservation ofangular momentum one should include the radiation reactionterm in the intrinsic part of the optical torque. Hence the opticaltorque, on an anisotropic subunit, can be written as

�(ri) = ri × F(ri) + 12 Re

{p(ri) × [

E(ri) + 23 ik3

0p(ri)]∗

+ m(ri) × [H(ri) + 2

3 ik30m(ri)

]∗}, (10)

where 23 ik3

0p(ri) and 23 ik3

0m(ri) are the electric and magneticreaction fields for a small polarizable particle, respectively[12].

III. RESULTS

Throughout this article we assume that the cloak is illumi-nated by a plane wave, traveling in the positive z direction withlinear or left circular polarization; see Fig. 1. In this sectionthe optical force and torque are normalized to to 4πε0|Einc|2.Hence optical forces and torques are given in squared meterand cubic meters, respectively.

FIG. 1. (Color online) Sketch of the geometry. A cloak with outerradius twice the inner radius. Illumination by a plane wave travelingin the positive z direction and with either linear polarization (electricfield along the x axis) or left circular polarization.

A. Optical force and torque on a cloak without material losses

To get a sense of the influence of the discrete geometry ofthe cloak on optical forces, we first study the optical force on alossless spherical invisibility cloak illuminated by a plane waveat wavelength λ. For this type of cloak the relative permittivityand permeability can be written as [18]

ε(r) = μ(r) = a

a − b

(I − 2br − b2

r4r ⊗ r

), (11)

where the cloak is centered on the origin with inner radius b andouter radius a chosen such that a/b = 2. We recall that for anideal cloak (made of continuous, nondispersive, and nonlossymaterials) the incident wave experiences no scattering, andthere would obviously be no net optical force on the cloak, aswas shown recently in Ref. [1].

1. Influence of the discretization on the force experiencedby a discrete cloak

We now use the DDA to study the influence of the size of thediscretization, or equivalently of the number of subunits, onthe optical force experienced by the cloak. In Fig. 2(a) the netoptical force on the cloak is plotted versus the number of layersof discretization of the cloak (Nl = 2a/d, where d is the size ofthe discretization mesh) for three different radii of the cloak. AsNl increases the net optical force decreases to zero; i.e., the dis-crete cloak tends toward an idealized, continuous cloak. How-ever, it is interesting to note that even with a coarse discretiza-tion the optical cloaking effect is noticeable. To better appre-ciate this and quantify the cloaking effect, consider the opticalforce that a homogeneous dielectric sphere of the same size asthe cloak but with ε = 2.25 and μ = 1 would experience (seeTable I; notice that the computation of the optical force experi-enced by the sphere is done exactly using a Mie series). For thetwo smallest radii, a discrete spherical cloak with only Nl = 16

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20 40 60 80 10010

−22

10−20

10−18

10−16

10−14

Nl

F/(

4π ε

0 |Ein

c |2 ) [m

2 ]

a=λa=λ/2a=λ/4

20 40 60 80 10010

−30

10−28

10−26

10−24

10−22

Nl

Γ/(4

π ε 0 |E

inc |2 )

[m3 ]

a=λa=λ/2a=λ/4

(a)

(b)

FIG. 2. (Color online) Optical force and torque on a discrete cloakof outer radius a (inner radius b = a/2). Solid line a = λ, dashedline a = λ/2 and in dash-dotted line a = λ/4. (a) Optical force onthe cloak due to a linearly polarized plane wave versus the numberof layer Nl . (b) Optical torque on the cloak due to a left circularlypolarized plane wave versus the number of discretization layer Nl .

discretization layers can reduce the optical force by morethan three orders of magnitude compared to the homogeneousdielectric sphere. This strong decreases of the electromagneticforce shows that a discrete cloak with this level of discretizationis quite effective as a shield against electromagnetic forces. Fora = λ the larger radius means that we must discretize the cloakmore finely (Nl = 40) in order to observe the same attenuationof the force compared to the homogeneous sphere (notice thatNl = 16 and Nl = 40 correspond to 1896 and 29 328 subunitsto represent the cloak, respectively).

For a linearly polarized incident plane wave the invisibilitycloak experiences a zero optical torque, but this is no longertrue for a circularly polarized incident wave [see Fig. 2(b)].In that case we see that for small values of Nl the cloakexperiences an optical torque resulting from the anisotropyof the permittivity [19] and permeability of the cloak. WhenNl increases the optical torque vanishes as, once again, thediscretized cloak tends toward an ideal (continuous) cloak.

TABLE I. Comparison between the optical force experienced bya homogeneous dielectric sphere (ε = 2.25, μ = 1) and a sphericalcloak for different radii and level of discretization for the cloak.

Radius a = λ/4 a = λ/2 a = λ

Homogeneous sphere 1.1 × 10−15 1.1 × 10−14 4.4 × 10−14

Cloak Nl = 16 1.2 × 10−19 2.8 × 10−18 2.0 × 10−15

Cloak Nl = 40 3.6 × 10−21 1.3 × 10−19 7.0 × 10−17

2. Density of optical force and torque inside the cloak

It is interesting to look at the optical force (torque) on eachelement of the composite structure as it gives us an insight inthe mechanical stress inside the cloak. Moreover, if the opticalforce (torque) experienced by each element forming the cloakis normalized to its volume, we obtain the density of force(torque) inside the cloak. In Figs. 3(a)–3(c) [Figs. 3(d)–3(f)] weplot the density of force (torque) as a vector field with Nl = 23for a = λ/4 and a = λ/2 and Nl = 41 for a = λ. On the samegraphs [Figs. 3(a)–3(f)] the color scale represents the densityof electromagnetic energy. Figure 3 highlights the fact that thedensity of energy is uniform outside the cloak and negligiblewithin the cloak. Notice that the energy density is largest nearthe outer boundary of the cloak. Thus, even if the net opticalforce experienced by the cloak vanishes as the metamaterialelements become smaller, there remains nonetheless a nonzerodensity of optical force (DOF), which means that the cloakis subject to mechanical stress, a result in agreement withthe study of an ideal, continuous cloak presented in Ref. [1].Figures 3(a)–3(c) show that for all three sizes of cloaks, themechanical stress “stretches” the cloak in the z direction and“compresses” it in the x direction. Note that the amount ofstress decreases with the radius of the cloak, which can beunderstood intuitively in terms of the curvature imposed bythe cloak on the incident wave.

For the three cloaks the maximum of DOF is more orless the same, i.e., about 2×107 Nm−3. One can see that thestress is stronger close to the interior edge (particularly forthe larger cloak) where the density of energy vanishes. Thus,although the density of energy and the density of force areboth quadratic forms in the fields, they behave differently.Regarding the density of optical torque, Figs. 3(d)–3(f) showsthat it vanishes overall for the cloak, but there remains anonzero density of optical torque inside the cloak. Notice thatone cannot obtain the density of optical torque directly fromFigs. 3(a)–3(c) with r × F as one should add the intrinsic partof the optical torque Eq. (10).

3. Optical force on a particle inside the cloak

In this section we study the optical force experiencedby a minute test particle of radius λ/72 located within thediscrete cloak. We emphasize that the presence of the testparticle is self-consistently taken into account. This meansthat the electromagnetic coupling between the test particleand the cloak is taken into account to compute the opticalforce experienced by the test particle. Figure 4(a) shows thez component of the optical force experienced by the particle(either lossless or absorbing) versus its position along the z axisinside the cloak (x = y = 0). For both particles, we see thatwhen the particle is close to the inner boundary of the cloakit experiences a repulsive force that pushes it away from theboundary. Note, however, that this is not a general behavior.If we look at the optical force experienced by a particle asit moves inside the cloak (not represented) we observe onecommon feature: The magnitude of the optical force is weakwhen the particle is far from the internal edge and increasesdramatically near the edge. As for the specific spatial profile ofthe force on the particle, it will depend on several parameterssuch as the polarizability of the particle and the size of the

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−0.2 0 0.2

−0.2

−0.1

0

0.1

0.2

z/λ

x/λ

107 0.02

0.04

0.06

0.08

0.1

0.12

−0.2 0 0.2

−0.2

−0.1

0

0.1

0.2

z/λ

x/λ

0.05 0.02

0.04

0.06

0.08

0.1

0.12

−0.5 0 0.5−0.5

0

0.5

z/λ

x/λ

107 0.02

0.04

0.06

0.08

0.1

0.12

−1 0 1−1

−0.5

0

0.5

1

z/λ

x/λ

107 0.02

0.04

0.06

0.08

0.1

0.12

−0.5 0 0.5−0.5

0

0.5

z/λ

x/λ

0.05 0.02

0.04

0.06

0.08

0.1

0.12

−1 0 1−1

−0.5

0

0.5

1

z/λ

x/λ

0.20.02

0.04

0.06

0.08

0.1

0.12

k0

k0

incE

incE

(a) (b) (c)

(d)

Density of optical force

Density of optical torque(e) (f)

FIG. 3. (Color online) Cross sections in the y = 0 plane. Vector fields representing the density of force (first row: a, b, c) for a cloakilluminated by a linearly polarized plane wave and the density of optical torque (second row: d, e, f) on a cloak illuminated by a left circularlypolarized plane wave. A sketch of the polarization state of the incident wave is given on the left side of the figure. In all figures, the color scalerepresents the density of electromagnetic energy. The scale bar for the density of force or torque is given in the bottom left corner of eachfigure. (a) and (d) a = λ/4 and Nl = 23. (b) and (e) a = λ/2 and Nl = 23. (c) and (f) a = λ and Nl = 41.

cloak; the particle can be attracted toward or pushed awayfrom the internal edge of the cloak. However, we notice that,inside the cloak, the force on the absorbing particle is strongerthan the one on the lossless particle, irrespective of the locationinside the cloak. This is due to a stronger in radiation pressureon the absorbing particle [20,21].

To better see the influence of the cloak on the force experi-enced by the particle we plot in Fig. 4(b) the force normalizedto the force the particle would experience in the absence ofthe cloak, i.e., in free space. For the absorbing particle theresult is as expected: The cloak hides the particle, andthe optical force experienced by the particle decreases whenthe particle is within the cloak. This, however, is not the casefor the lossless particle as the optical force experienced bythe particle increases when it is located within the cloak. Thissomewhat counterintuitive effect is due to the difference be-tween radiation pressure and gradient force. When the particleis inside the cloak, the electromagnetic field is weak (at the cen-ter of the cloak the density of energy is less than 0.4% of that ofthe incident field), hence the radiation pressure is weak. How-ever, the spatial gradient of the fields is not negligible insidethe cloak, particularly close to the interior edge. The particletherefore experiences a gradient force (which is proportionalto the real part of the polarizability), and the optical force ex-perienced by the particle is stronger within the cloak (near theinterior edge), than in free space (no gradient force). Of course,at the center of the cloak, the total optical force experienced bythe lossless particle is indeed weaker than in free space as thegradient force vanishes there. When absorption is present wealso have a contribution to the optical force that is proportionalto the imaginary part of the polarizability. The combination of

gradient force and absorption results in, first, a shift betweenthe two curves’ [solid line for the lossless particle and dashedline for the absorbing particle in Fig. 4(a)] and, second, aweaker normalized force due to a stronger force on the absorb-ing particle in the free-space case [dashed line in Fig. 4(b)].

B. Optical force and torque on a dispersive cloak

So far we have neglected material absorption in the cloak soas to highlight the influence of the discrete geometry; however,it is impossible to build a metamaterial with the required opticalproperties over a wide frequency range without dispersionand absorption. Therefore, in this section we study theinfluence of dispersion and losses in the cloak by introducinga dispersion profile in the definition of the permittivity andpermeability tensors associated with the discrete subunitsforming the cloak. Assuming a Lorentz dispersion profile [12],which is commonly used to model the material dispersion ofmetamaterial structures, we define [22,23]

f (ω) = f∞ − F

ω2 + iω − ω2g

, (12)

where ωg and are the transition frequency and damping rate,respectively. Equation (12) satisfies causality and is Kramers-Kronig consistent [12,24]. The permittivity and permeabilitytensors defined in Eq. (11) are now multiplied by the functionf (ω) to account for material dispersion. Notice that if = 0and f (ω) �= 1, the cloak is no longer perfect although it is stilllossless. However, if is different from zero, then f (ω) has anonzero imaginary part and the optical cloak is lossy as a lossterm is introduced in both the permittivity and permeability.

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−0.2 −0.1 0 0.1 0.2

−2

0

2

4x 10

−22

z/ λ

F/(4

πε0|E

inc |2 )

ε=2.25ε=2.25+i

−0.2 −0.1 0 0.1 0.2−100

−60

−20

20

60

100

z/λ

Nor

mal

ized

opt

ical

for

ce

−0.01

−0.005

0

0.005

0.01

ε=2.25ε=2.25+i

(a)

(b)

FIG. 4. (Color online) Force on a minute particle of radius λ/72inside the cloak with a = λ/2, b = a/2, and Nl = 45. Losslessparticle: ε = 2.25; solid line and left vertical axis in (b). Absorbingparticle: ε = 2.25 + i; dashed line and right vertical axis in (b). (a) z

component of the optical force is plotted versus z for x = y = 0. (b)Same as (a), but the optical force is normalized to the optical forceexperienced by the same particle located in free space.

1. Dispersive discrete cloak: force

We choose the illumination wavelength λ0 = 2πc/ω0 andthe parameters of the dispersion profile such that f (ω0) = 1when = 0. This means that at wavelength λ0 the relativepermittivity and permeability of the cloak matches Eq. (11).With = 0 at λ = λ0 the cloak is lossless, but if thewavelength of illumination λ is changed from λ0, the cloakis lossy. Figure 5 shows the optical force on a cloak with outerradius a = λ0/4 versus the wavelength of illumination whenthe cloak is dispersive. The solid line curve without circles(dashed line curve without circles) gives the spectrum of theforce for = 0 for an illumination frequency either belowthe transition frequency ωg (ωg = 3ω0, F = 90ω2

0) or abovethe transition frequency ωg (ωg = ω0/10, F = ω2

0/3) [theinset in the bottom left (right) corner shows f (ω)]. Whenλ = λ0 the cloak is an ideal discrete cloak, hence the weakoptical force. However, due to the dispersive nature of thecloak, if the wavelength of illumination is moved away fromλ0, the optical force on the cloak increases by several orders ofmagnitude. If we introduce a small amount of absorption, i.e., = 10−3, the real part of f (ω) is not noticeably changed overthe frequency range of interest and Im[f (ω)] ≈ 10−3. Figure 5(plots with circles) shows that the introduction of losses entails

0.8 1 1.2

10−20

10−18

10−16

10−14

λ/λ0

F/(

4πε 0|E

inc |2 )

[m2 ]

0.8 1 1.20.6

0.8

1

1.2

λ/λ0

f(ω

)

0.8 1 1.20

1

2

3

λ/λ0

f(ω

)

FIG. 5. (Color online) Optical force on a dispersive cloak versuswavelength for ωg = 3ω0 and = 0 (solid line), ωg = 3ω0 and =0.001 (solid line with circles), ωg = ω0/10 and = 0 (dashed line),ωg = ω0/10 and = 0.001 (dashed line with circles). The left (right)inset is f (ω) when ωg = 3ω0 (ωg = ω0/10) and = 0.

an increase of the optical force around λ = λ0, revealing aweakening of the cloaking mechanism by absorption.

2. Dispersive discrete cloak: torque

The optical torque on the cloak, for the strongly dispersivecase ωg = 3ω0, is presented in Fig. 6. The cloak is illuminatedwith a circularly polarized plane wave. One can see thatthe introduction of a small amount of material absorption( = 0.001) increases the torque significantly. Note, however,the different effect material absorption has on the optical torqueand force. For the optical force the magnitude of the force isaltered significantly by material absorption only in a narrowspectral region around λ0. On the other hand, for the opticaltorque, the lossless ( = 0) and lossy ( = 0.001) casesare noticeably different over a wider range of frequencies.This is due to the fact that the optical torque consists oftwo contributions: one from absorption and one from the

0.8 1 1.210

−30

10−28

10−26

10−24

10−22

λ/λ0

Γ/(4

πε0|E

inc |2 )

[m3 ]

FIG. 6. (Color online) Optical torque versus the wavelength on adispersive cloak when ωg = 3ω0, = 0 (solid line), and = 0.001(dashed line). Circles: absorbing cross section of the cloak normalizedto 8π 2/λ.

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CHAUMET, RAHMANI, ZOLLA, AND NICOLET PHYSICAL REVIEW E 85, 056602 (2012)

anisotropy of the permeability and permittivity tensors. InFig. 6 we plot, using circles (no line), the absorbing crosssection of the cloak normalized to 8π2/λ. Clearly the cross-section spectrum follows very closely the spectral evolution ofthe optical torque, showing that, as far as the optical torque isconcerned, the cloak behaves like an absorbing homogeneoussphere [25], i.e., the optical torque is proportional to theabsorbing cross section.

3. Effect of the dispersion on a hidden particle within the cloak

With invisibility cloaks the focus is traditionally on theconcealment of an object from electromagnetic probes. Inparticular, a cloak is expected to induce very little (none inthe ideal case) scattering of electromagnetic waves. However,as we have seen, the scattering of waves by the cloak is onlyhalf the story. The electromagnetic probe can affect the cloakmechanically through optical forces and torques. Therefore,it would be interesting to assess the ability of a realisticcloak (with discrete geometry and material dispersion andabsorption) to shield an object from optical forces. To this endwe introduce a minute test particle at the center of the cloak andstudy how well the particle is protected from optical forces.The computation of the force experienced by the test particleis done self-consistently. Figure 7(a) shows the spectrum of

0.8 1 1.2

10−2

100

102

λ/λ0

Nor

mal

ized

opt

ical

for

ce

10−26

10−24

10−22

λ /λ0

F/(

4πε 0|E

inc |2 ) [

m2 ]

(a)

(b) 0.8 1 1.2

FIG. 7. (Color online) We consider both a lossless and a lossycloak for two dispersion profiles: ωg = 3ω0, = 0 (solid line), and = 0.001 (solid line with circles); ωg = ω0/10, = 0 (dashed line),and = 0.001 (dashed line with circles). (a) Spectrum of opticalforce experienced by a test lossless particle (ε = 2.25 and μ = 1)located at the center of a dispersive cloak. (b) The force is normalizedto the optical force the particle would experience in free space (in theabsence of the cloak).

the optical force experienced by a lossless dielectric particle(ε = 2.25 and μ = 1) at the center of the cloak. In Fig. 7(b)the optical force is normalized to the optical force the particlewould experience in the absence of the cloak. We considertwo dispersion profile parameters: ωg = 3ω0 (solid line) andωg = ω0/10 (dashed line) with = 0. It can be seen that theoptical force increases dramatically when the wavelength ofillumination moves away from λ0 [recall that the dispersionprofile of the cloak is chosen such that f (ω0) = 1 in the losslesscase] due to the deterioration of the cloaking mechanism: Ifthe field inside the cloak is no longer negligible, the opticalforce on the test particle increases. This is particularly obviousin Fig. 7(b), where the normalized optical force is larger thanone, meaning that the sphere experiences a stronger opticalforce inside the cloak than the force it would experience infree space. Also, the test particle would not remain at thecenter of the cloak and would move to another equilibriumposition or hit the edge of the cloak. Notice that if a weakabsorption is added to the permittivity and permeability ofthe cloak ( = 0.001, curves with circles), the optical forcedoes not change significantly. If the concealed particle is itselfabsorbing, (ε = 2.25 + i and μ = 1), we see in Figs. 8(a) and8(b) that the normalized optical force remains close to thevalue obtained at λ0 and always less than one. This illustratesthe fact that with a leaky and dispersive cloak, the opticalforce experienced by an absorbing particle is lower within thecloak than in free space by two orders of magnitude in all therange of wavelength studied, hence the particle is protectedby the cloak. This illustrates the more or less intuitive factthat even with a leaky and dispersive cloak (i.e., a nonideal

0.8 1 1.210

−23

10−22

10−21

λ/λ0

F/(

4πε 0|E

inc |2 )

[m2 ]

0.8 1 1.210

−4

10−3

10−2

10−1

λ/λ0

Nor

mal

ized

Opt

ical

for

ce

(a)

(b)

FIG. 8. (Color online) Same caption as in Fig. 7, but the testparticle is an absorbing particle: ε = 2.25 + i and μ = 1.

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ELECTROMAGNETIC FORCES ON A DISCRETE . . . PHYSICAL REVIEW E 85, 056602 (2012)

cloak), an absorbing particle is sheltered by a cloak in contrastto a lossless particle even if the net force experienced by anabsorbing particle were larger than that of a lossless particle.

IV. CONCLUSION

We have studied the influence of the discrete geometry ofa spherical invisibility cloak on the time-averaged force andtorque experienced by the cloak in the time-harmonic regime.We found that a significant cloaking effect can be achievedeven with a relatively coarse discrete geometry. We alsoconsidered the effect of material dispersion and absorption andshowed that a small amount of losses can lead to a significant

increase in the optical force and torque experienced by thecloak. The capacity of a cloak to shield an object has beeninvestigated by considering a test particle inside the cloak, andcomparing the optical force on the particle to the force it wouldexperience in free space (in the absence of the cloak). Whileeven an imperfect (dispersive and lossy) cloak can reduce theoptical force on the particle, the “mechanical shielding” ofthe particle will greatly depend on the particle’s locationinside the cloak and whether the particle is lossy. In particular,because optical forces depend not only on the magnitude of thefields but also on their gradients, near the the internal boundaryof the cloak, the optical force can exceed what it would be inthe absence of the cloak.

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